Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-27T23:14:23.056Z Has data issue: false hasContentIssue false
  • English
  • Français

Application of the Level-Set Model with Constraints in Image Segmentation

Part of: Parabolic equations and systems Numerical methods Global differential geometry Equilibrium (steady-state) problems

Published online by Cambridge University Press:  15 February 2016

Vladimír Klement ,
Tomáš Oberhuber  and
Daniel Ševčovič
Vladimír Klement
Affiliation:
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Praha 2, 120 00, Czech Republic
Tomáš Oberhuber
Affiliation:
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Praha 2, 120 00, Czech Republic
Daniel Ševčovič*
Affiliation:
Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava, Slovakia
*
*Corresponding author. Email addresses: [email protected] (T. Oberhuber), [email protected] (V. Klement), [email protected] (D. Ševčovič)
Get access

Abstract

We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper and lower constraints prescribed for the level-set function. Constraints have the same conceptual meaning as initial seeds of the popular graph-cuts based methods for image segmentation. A numerical approximation scheme is based on the complementary-finite volumes method combined with the Projected successive over-relaxation method adopted for solving constrained linear complementarity problems. The advantage of the constrained level-set method is demonstrated on several artificial images as well as on cardiac MRI data.

Keywords

linear complementarityimage processingsegmentationlevel-set methodprojected successive over-relaxation method

MSC classification

Secondary: 90C33: Complementarity and equilibrium problems and variational inequalities (finite dimensions) 35K55: Nonlinear parabolic equations 35K52: Initial-boundary value problems for higher-order parabolic systems 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 74S10: Finite volume methods 74G15: Numerical approximation of solutions
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 1 , February 2016 , pp. 147 - 168
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Beneš, M., Kimura, M., Pauš, P., Ševčovič, D., Tsujikawa, T., Yazaki, S., Application of a curvature adjusted method in image segmentation, Bulletin of Inst. of Mathematics, Academia Sinica, New Series 3 (2008), 509–524.Google Scholar
[2]Beneš, M., Máca, R., Application of a degenerate diffusion method in 3D medical image processing, Proceedings of Agoritmy 2012, Handlovičová, A., Minarechová, Z. and Ševčovič, D. (ed.), (2012), 416–426.Google Scholar
[3]Boykov, Y., Funka-Lea, G., Graph cuts and efficient n-d image segmentation, International Journal of Computer Vision 70 (2006), 109–131.CrossRefGoogle Scholar
[4]Boykov, Y., Jolly, M. P., Interactive graph cuts for optimal boundary & region segmentation of objects in n-D images. In: Proceedings of International Conference on Computer Vision, Vancouver, Canada, July 2001, Vol. 1, 105–112, 2001.Google Scholar
[5]Boykov, Y., Kolmogorov, V, An experimental comparison of min-cut/max- flow algorithms for energy minimization in vision, Pattern Analysis and Machine Intelligence 26 (2004), 1124–1137.CrossRefGoogle ScholarPubMed
[6]Brézis, H., Problèmes unilatéraux, J. Math. Pures Appl. 51 (1972), 1–168.Google Scholar
[7]Caselles, V., Catté, F., Coll, T., Dibos, F., A geometric model for active contours in image processing, Numerische Mathematik 66 (1993), 1–31.CrossRefGoogle Scholar
[8]Caselles, V., R. Kimmel, , Sapiro, G., Geodesic active contours, in: Proceedings International Conference on Computer Vision' 95, Boston, pp. 694–699.Google Scholar
[9]Chan, T., Zhu, W., Level set based shape prior segmentation, in: Computer Vision and Pattern Recognition, IEEE Computer Society Conference on, 2, (2005), 1164–1170.Google Scholar
[10]Chung, G., Vese, L. A., Image segmentation using a multilayer level-set approach, Computing and Visualization in Science, 12(6) (2009), 267–285.CrossRefGoogle Scholar
[11]Cremers, D., Rousson, M., Deriche, R., A Review of Statistical Approaches to Level Set Segmentation: Integrating Color, Texture, Motion and Shape, Internationa Journal of Computer Vision, 72(2) (2007), 195–215.CrossRefGoogle Scholar
[12]Droske, M., Rumpf, M., A level set formulation for Willmore flow, Interfaces Free Boundaries, 6 (3) (2004), 361–378.CrossRefGoogle Scholar
[13]Elliott, C. M., Ockendon, J. R., Weak and Variational Methods for Moving Boundary Problems, Vol. 59, Research Notes in Mathematics, Pitman, Boston, Mass., 1982.Google Scholar
[14]Gurholt, T. P., Xuecheng, Tai, 3D Multiphase Piecewise Constant Level Set Method Based on Graph Cut Minimization, Numer. Math. Theor. Meth. Appl., 2 (2009), 403–420.Google Scholar
[15]Handlovičová, A., Mikula, K., Sgallari, F., Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution, Numerische Mathematik 93 (2003), 675–695.CrossRefGoogle Scholar
[16]He, L., Peng, Z., Everding, B., Wang, X., Han, Ch. Y., Weiss, K. L., Wee, W. G., A comparative study of deformable contour methods on medical image segmentation, Image and Vision Computing 26 (2008), 141–163.CrossRefGoogle Scholar
[17]Jiang, Y., Wang, M., Xu, H., A Survey for Region-based Level Set Image Segmentation, in: IEEE Proceedings of 11th International Symposium on Distributed Computing and Applications to Business, Engineering & Science (DCABES), 19-22 Oct. 2012, 413–416.CrossRefGoogle Scholar
[18]Kass, M., Witkin, A., Terzopoulos, D., Snakes: Active contour models, International Journal of Computer Vision 1 (1987), 321–331.CrossRefGoogle Scholar
[19]Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A., Jr., Conformal curvature flows: from phase transitions to active vision, Arch. Rational Mech. Anal. 134 (1996), 275–301.CrossRefGoogle Scholar
[20]Loucký, J., Oberhuber, T., Graph cuts in segmentation of a left ventricle from MRI data, Proceedings of Czech-Japanese Seminar in Applied Mathematics 2010, ed. Beneš, M., Kimura, M., Yazaki, S., COE Lecture Note, 36 (2012), 46–54.Google Scholar
[21]Mangasarian, O. L., Solution of symmetric linear complementarity problems by iterative methods, J. Optimization Theory Applicat. 22 (1977), 465–485.CrossRefGoogle Scholar
[22]Mikula, K., Sarti, A., Parallel co-volume subjective surface method for 3D medical image segmentation, in: Parametric and Geometric Deformable Models: An application in Bio-materials and Medical Imagery, Volume-II, Springer Publishers, (Eds. Jasjit Suri, S. and Farag, Aly), 2007, pp. 123–160.CrossRefGoogle Scholar
[23]Mikula, K., Ševčovič, D., Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Vis. Sci. 6 (2004), 211–225.CrossRefGoogle Scholar
[24]Oberhuber, T., Complementary finite volume scheme for the anisotropic surface diffusion flow, in: Proceedings of Algoritmy 2009, (2009), 153–164.Google Scholar
[25]Geometric Level Set Methods in Imaging, Vision, and Graphics, Osher, S., Paragios, N. (Eds.), Springer, New-York, 2003.Google Scholar
[26]Paragios, N., Deriche, R., Geodesic active contours and level sets for the detection and tracking of moving objects, in: Pattern Analysis and Machine Intelligence, IEEE Transactions on 22,(3) (2000), 266–280.Google Scholar
[27]Quarteroni, A., Sacco, R., Saleri, F., Numerical Mathematics, Springer, 2000Google Scholar
[28]Sethian, J. A., Level Set Methods, Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, New York, 1996.Google Scholar